How could AI solve planet's major problems?
There's indeed a lot of research work being done in this field.

Here are a few ways in which AI can help/is helping in fighting major problems:

Climate change


Identify deforestation and the rate at which it's happening using computer vision and help in fighting back based on how critical the rate is.The World Resources Institute had entered into a partnership with Orbital Insight on this.


Protecting Nature


Researchers and institutes are using computer vision to accurately identify total remaining members of extremely threatened species to observe their behavior and patterns that can help in protecting them, keep track and monitor at an individual level.
Environmental/ Research agencies are trying to detect sounds from ocean audio from which they can identify inhabiting fish and route the ships away from threatened habitats.


Energy


Optimise energy usage based on consumption in big companies to save energy for public use and in turn reduce energy overall demands on the grid.
Predict the demand for energy and ensure continuous supply matching the demand. For example, Google's Deepmind is in talks National Grid of UK on this.


Of course, these ideas might look too tiny or too broad at first sight, but I feel taking one step at a time and then quickly expanding the ideas into wider areas would be the way to go, like how DeepMind is doing it in the area of energy. Other areas like education, transportation, and healthcare are seeing lots of activity with the involvement of AI too.

For further reading, here's what World Economic Forum thinks about how AI can help solve world's problems.